QuestionBruce has 6 times more 10 cent stamps than 5 cent stamps, totaling 72 stamps worth \$8.40. Find the quantity of each type.

Studdy Solution

STEP 1

Assumptions1. The number of10 cent stamps is six times the number of5 cent stamps. . The total number of stamps is72.
3. The total value of all stamps is $8.40.
4. We are asked to find the number of each type of stamp.

STEP 2

Let's denote the number of5 cent stamps as $x$ , the number of10 cent stamps as $6x$ , and the number of20 cent stamps as $y$ .

STEP 3

We know that the total number of stamps is72. Therefore, we can write the equation $x +6x + y =72$

STEP 4

implify the equation to get $7x + y =72$

STEP 5

We also know that the total value of all stamps is $8.40 or840 cents. Therefore, we can write the equation$ $5x +10(x) +20y =840$ $

STEP 6

implify the equation to get $60x +20y =840$

STEP 7

We can simplify this equation further by dividing all terms by20 $3x + y =42$

STEP 8

Now we have a system of two equations1. $7x + y =72$
2. $3x + y =42$

STEP 9

Subtract the second equation from the first to find the value of $x$ $7x -3x =72 -42$

STEP 10

olve the equation to find the value of $x$ $4x =30$ $x =30 /4 =7.5$

STEP 11

Since the number of stamps must be an integer, there seems to be a mistake. Let's go back and check our equations. We made a mistake in5. The correct equation should be $5x +10(6x) +20y =840$

STEP 12

implify the equation to get $65x +20y =840$

STEP 13

We can simplify this equation further by dividing all terms by5 $13x +y =168$

STEP 14

Now we have a system of two equations. $7x + y =72$
2. $13x +4y =168$

STEP 15

To solve this system, we can multiply the first equation by4 and the second equation by $28x +4y =288$ $13x +4y =168$

STEP 16

Subtract the second equation from the first to find the value of $x$ $28x -13x =288 -168$

STEP 17

olve the equation to find the value of $x$ $15x =120$ $x =120 /15 =$

STEP 18

Substitute $x =8$ into the first equation of our system to find the value of $y$ $7x + y =72$ $7(8) + y =72$

STEP 19

olve the equation to find the value of $y$ $56 + y =72$ $y =72 -56 =16$ Bruce has8 five-cent stamps,48 ten-cent stamps, and16 twenty-cent stamps.

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